PCCP PAPER. Exploring non-adiabatic approximations to the exchange correlation functional of TDDFT. I. Introduction. Johanna I. Fuks, Neepa T.

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1 PAPER View Journal View Issue ite this: Phys. hem. hem. Phys., 2018, 20, Received 22nd June 2018, Accepted 26th September 2018 DOI: /c8cp03957g rsc.li/pccp I. Introduction Exploring non-adiabatic approximations to the exchange correlation functional of TDDFT Johanna I. Fuks, Neepa T. Maitra Much effort has been put into the development of more accurate and reliable approximations to the ground state (gs) exchange correlation (xc) functional of Density Functional Theory (DFT). 1 3 In order to make use of these approximations in the simulation of dynamical processes one invokes the adiabatic approximation: one utilizes a gs xc functional in place of the time-dependent one, and so implicitly assumes the many-body effects in the system are those of a gs even if the system is itself not near a gs. Memory effects arising from the dependence of the xc functional on the history of the density, and on the initial states, are completely absent in these approximations. Adiabatic Time-dependent Density Functional Theory (TDDFT) is widely and successfully used to simulate excitation spectra and response properties in finite systems and solids that are initially in the gs 4 7 and probe the perturbative regime, where the system is only slightly disturbed from its gs. The adiabatic approximation becomes less justified as we move away from the gs, as is the case when the perturbation is strong or resonant or when we start thesimulationinanexcited or arbitrary superposition state. In such situations, which have become of increasing interest in the past decade due to the a Departamento de Física and IFIBA, FEN, Universidad de Buenos Aires, iudad Universitaria, Pab. I, 1428EHA iudad de Buenos Aires, Argentina. johannafuks@gmail.com b Department of Physics and Astronomy, Hunter ollege and the Graduate enter of the ity University of New York, 695 Park Avenue, New York, New York 10065, USA c Max Planck Institute for the Structure and Dynamics of Matter and enter for Free-Electron Laser Science, Luruper haussee 149, Hamburg, Germany Electronic supplementary information (ESI) available. See DOI: /c8cp03957g * a Lionel Lacombe, b Søren E. B. Nielsen c and b A decomposition of the exact exchange correlation potential of time-dependent density functional theory into an interaction component and a kinetic component offers a new starting point for nonadiabatic approximations. The components are expressed in terms of the exchange correlation hole and the difference between the one-body density matrix of the interacting and Kohn Sham systems, which must be approximated in terms of quantities accessible from the Kohn Sham evolution. We explore several preliminary approximations, evaluate their fulfillment of known exact conditions, and test their performance on simple model systems for which available exact solutions indicate the significance of going beyond the adiabatic approximation. development of experimental techniques and especially the growing interest in attosecond physics (e.g. ref. 8), the adiabatic approximation is at least questionable. How severely the adiabatic propagation deviates from the exact dynamics will depend on the system and the particular dynamics, and also on what the observables of interest are. For example, highly non-perturbative processes such as resonant dynamics and long-range charge transfer have been shown to be particularly challenging for adiabatic TDDFT 9 22 andeveninthecaseoffield-freeevolution of a non-stationary state the exact xc functional develops nontrivial features that influence the dynamics but that adiabatic functionals cannot capture On the other hand, for example, the adiabatic approximations give good predictions for photoemission spectra and photo-angular distributions in clusters. 26 Given the lack of alternative methods to simulate electrondynamics in medium to large systems and the expanding range of time- and space-resolved experiments involving nonperturbative dynamics it is imperative to explore the capability of TDDFT beyond the adiabatic approximation. There have been only a few proposals for non-adiabatic xc functionals, but they are not commonly used for a variety of reasons including that they spuriously damp finite systems. Functionals that are explicit orbital-functionals also incorporate memory (e.g. time-dependent exact exchange 30,31 ) but the orbital-dependence of these commonly only involves exchange which is inadequate to capture the most significant non-adiabatic effects. In this work we present a new starting point for developing non-adiabatic approximations based on an exact decomposition of the xc functional. The exact decomposition, which is reviewed in Section II, expresses the exact xc potential, in terms of an interaction component that depends on the xc hole of the interacting system, and a kinetic component that depends on This journal is the Owner Societies 2018 Phys. hem. hem. Phys., 2018, 20,

2 the difference between the one-body density matrices of the interacting and Kohn Sham systems. 24,32 The idea is to approximate these components in terms of quantities that are accessible from the evolving Kohn Sham system and/or from the initial many-body wavefunction, and some preliminary approximations areproposedinsectionii.1.insection III we analyze these in termsofthefulfillmentofexactconditionsandinsectionivwe test their performance against the exact solution for some model systems. Finally, in Section V we conclude. II. Decomposition of v xc into interaction and kinetic potentials Equating the equation of motion for the second time-derivative of the one-body density, n (r,t), in the physical system, with that in the Kohn Sham (KS) system n S(r,t), yields the following equation 24,32 for v xc (r,t): rnrv xc Þ¼r 1 4 r0 rþ r 2 r 02 r1 r 0 ;r;tþ r 1;S r 0 ;r;tþ þnr;tþ n xc r;r 0 ;tþrwjr 0 rjþd 3 r ; 0 r 0 ¼r (1) where r 1 r 0 ; r; tþ ¼N P Ð d 3 r 2...d 3 r N r 0 s 1 ; r 2 s 2...r N s N ; tþ s 1...s N rs 1 ; r 2 s 2...r N s N ; tþ is the spin-summed one-body densitymatrix (1RDM) of the true system of electrons with two-body interaction potential w( r r 0 ), r 1,S (r 0,r,t) is the 1RDM for the Kohn Sham system, and n xc (r,r 0,t) is the xc hole, defined via the pair density through Pr;r 0 ;tþ¼nn 1Þ P Ð j r 0 s 1 ; s 1...s N rs 2 ;r 3 s 3...r N s N ;tþj 2 d 3 r 3...d 3 r N ¼ nr;tþnr 0 ;tþþn xc r;r 0 ;tþþ. The notation r means the gradient with respect to r while r 0 means the gradient with respect to r 0. Eqn (1), with mixed separated boundary conditions (i.e. involving the potential and its gradient at each boundary point separately) has the form of a nonhomogeneous Sturm Liouville problem for v xc (r,t). We can decompose this potential into an interaction (W) and a kinetic (T) component, v xc (r,t) =v W xc(r,t) +v T (r,t) (2) where the interaction part (W) has an explicit dependence on the xc hole, rv W xc ¼ n xc r; r 0 ; tþrwjr 0 rjþd 3 r 0 þ 1 nr; tþ ra 1r; tþ; (3) and the kinetic (T) component, rv T ¼ r0 rþ r 2 r 02 Dr1 r 0 ; r; t 4nr; tþ þ 1 nr; tþ ra 2r; tþ Þj r 0 ¼r has an explicit dependence on spatial gradients of Dr 1 (r 0,r,t) = r 1 (r 0,r,t) r 1,S (r 0,r,t) evaluated at r = r 0. We note that, despite only their curl appearing in the equations, both a i (r,t) are fixed (4) uniquely by the fact that the right hand sides of eqn (3) and (4) must be longitudinal (in the sense of Helmholtz decomposition) and satisfy the boundary conditions. Numerically, they can be found by solving the Sturm Liouville equation directly for the potential, and taking the gradient of the potential. In Appendix (B), we will show that the corresponding terms are needed for rv xc to satisfy the generalized translational invariance condition. In one-dimension, the ra i (r,t) termsvanish. An analogous decomposition of the exact gs xc potential has provided insight into features of the gs xc potential, while ref. 24 explored how non-adiabatic features of the timedependent xc potential manifest themselves in each of these terms. For example, it was found that the dynamical step features found in ref. 15, 23 and 37 predominantly appear in v T, while v W xc appears generally much smoother. In this work we take the first steps to move the decomposition beyond being simply a tool to analyze the exact potential to being a foundation on which to build non-adiabatic functional approximations. By virtue of the Runge Gross theorem 4 the exact xc potential is a functional of the density n(r,t), the initial many-body state (r 1 s 1...r N s N ;0), and the initial Kohn Sham state F(r 1 s 1...r N s N ;0): v xc (r,t) =v xc [n;(0),f(0)](r,t). But eqn (3) and (4) depend explicitly on instantaneous many-body quantities such as the xc hole density n xc (r,r 0,t) andthe1rdm r 1 (r 0,r,t), as well as the Kohn Sham 1RDM r 1,S (r 0,r,t), and we do not know the functional form of these in terms of the density and the initial wavefunctions. Operating TDDFT via the timedependent Kohn Sham equations offers a partial simplification: one may search for explicit functionals of the time-evolving Kohn Sham orbitals, which are themselves implicit functionals of n(r,t) and F(0), and, by putting more information in the variables that the functional depends on in this way, one hopes that the functional form itself can be simpler. For example, the Kohn Sham 1RDM is directly accessible in terms of the instantaneous Kohn Sham orbitals, r 1;S r 0 ; r; tþ ¼ P f i r 0 ; tþf i r; tþ,and i;occ: has implicit dependence on n(r,t 0 o t) and F(0), including memory. The terms depending on n xc and r 1 are however still unknown as functionals of f i (r,t). In the next section we propose some preliminary approximations to v W xc and v T in terms of the instantaneous Kohn Sham orbitals (orbital functionals) and many-body quantities evaluated at initial time (frozen approximations), that can be used in practical propagation schemes. A. Approximations arising from the exact decomposition 1. v S xc. Here we replace all the quantities from the interacting wavefunction with Kohn Sham quantities on the RHS of eqn (1), defining a single-particle approximation that we denote by v S xc. It is then immediately evident that the kinetic term goes to zero, and such an approximation is an approximation of the interaction The initial many-body quantities are obtained from an interacting many-body computation at the initial time, consistent with the exact functionals being dependent on the initial many-body wavefunction Phys. hem. hem. Phys., 2018, 20, This journal is the Owner Societies 2018

3 component v W xc, making the following replacement in the RHS of eqn (3): v S xc: n xc - n S xc (5) where the KS xc hole n S xc is defined via the KS pair density P S r 0 ; r; tþ NN 1Þ Ð jfr 0 ; r; r 3...r N ; tþj 2 dr 3...dr N ¼ nr; tþ nr 0 ; tþ þn S xc r; r0 ; tþ. This approximation was first presented in ref. 25, where the shape and size of v S xc were shown to be close to v W xc for a particular dynamics of a model system. This was field-free dynamics of an initial interacting state prepared in a 50 : 50 superposition of a ground and first excitedstate. Ref. 25 found that for a wide range of choices of the initial Kohn Sham state F(0), the approximate potential v S xc remained very close to v W xc throughout the dynamics. The approximation was also tested for time-resolved electron-atom scattering in ref. 38 and 39, where it captured the approach of the electron to the target far better than the conventional approximations, although, like them, also ultimately severely underestimated the scattering probability. The integral in eqn (3) tends to smooth out details of the xc hole somewhat, making v W xc less sensitive to the approximation for the xc hole than v T is to the approximation made for Dr 1 in eqn (4) as we will see shortly. From its definition, v S xc is an orbital functional, and thus generally is non-adiabatic in that it depends implicitly on the history of the density and on the Kohn Sham initial state. When a Slater determinant is chosen for the Kohn Sham initial state, it is probable that v S xc reduces to time-dependent exactexchange (TDEXX). For two electrons this follows directly from the formulae, but for more electrons, it is not as straightforward to show. For a more general F(0), v S xc goes beyond exact-exchange, and contains a portion of correlation. 25 In either case, it has spatial- and time-non-local dependence on the density, and dependence on the KS initial state, through the dependence on the KS orbitals. Note that in performing eqn (5) and in all following approximations, the terms a i are adjusted to maintain that the RHS of eqn (3) and (4) are purely longitudinal. 2. v S xc + v T (0). We now address approximations for the kinetic term to be used in conjunction with v S xc. Our first approximation that has a non-zero contribution from the The reasoning goes as follows. TDEXX is defined via the time-dependent optimized effective potential, 30 which is derived from a stationary principle based on an action obtained from evaluating H iq/qt on a Slater determinant state, with H being the full interacting Hamiltonian. 30,31,40 If instead the exact interacting wavefunction was used, the procedure would yield the exact xc potential (and in general needs to be done on the Keldysh contour 40 ); TDEXX can be thought of as assuming F = is a Slater determinant in such a procedure. Now the force-balance equation for n which gives rise to eqn (1) also yields the exact xc potential, and v S xc results if we make the exact same approximation, F = = Slater determinant in this equation. The latter corresponds to v Loc x in ref. 9, which the authors compare against exchange-only-kli for some particular dynamics and find the two yield similar results. It is extremely difficult to show directly from the formulae whether the two expressions are equal, 41 but the argument above suggests they could be, or are close. kinetic part of the correlation potential is very simple: we freeze it to its initial value, that is, on the RHS of eqn (4) v T (0): Dr 1 (t) - Dr 1 (0); n(t) - n(0). (6) This is of a somewhat similar spirit to the frozen approximations that are popular in strong-field atomic and molecular physics, such as the single-active electron approximation 42 where it is assumed that only one electron is responsible for the dynamics while all the rest are frozen, such that if the approximation was 1-electron self-interaction free, the potential that the dynamic electron sees would be static. It is also similar to the instantaneous ground-state approximation of ref. 43 when any externally applied field is static. In addition to the non-adiabatic effects contained in v S xc this approximation also is dependent on the initial interacting and KS states through its dependence on Dr 1 (0). 3. v S xc þ vt;dr 1 0Þ. To thaw the frozen kinetic component somewhat, we consider now the next simplest approximation, which is to freeze only the 1RDM-difference in v T, i.e. v T;Dr 1 0Þ : Dr 1 tþ!dr 1 0Þ (7) while self-consistently updating the density in the denominator on the RHS of eqn (4). Like v S xc + v T (0), the approximation v S xc þ vt;dr 1 0Þ inherits a degree of non-adiabatic density-dependence and Kohn Sham initial-state dependence through the instantaneous orbital-dependence in v S xc, and a degree of both Kohn Sham and interacting initial-state dependence through the kinetic component. 4. v S xc þ vt;r 10Þ. Noting that r 1,S is accessible during the Kohn Sham evolution, we define a further frozen approximation where only the interacting 1RDM is fixed to its initial value, while both the Kohn Sham 1RDM and the density are self-consistently updated during the time-evolution: v T;r 10Þ : Dr 1 tþ! r 1 0Þ r 1;S tþ : (8) This approximation freezes the interacting 1RDM while the Kohn Sham 1RDM may vary greatly. In the next sections we will discuss the fulfillment of exact conditions and the performance in reproducing some model dynamics for each of the proposed approximations. III. Exact conditions Exact conditions have been a key instrument guiding the development of physically-inspired and robust gs density functionals that work reliably over a wide range of systems. 44,45 Except for the ones arising from the energy-minimization principle which cannot be extrapolated to the time-dependent case, many of the gs conditions should be also fulfilled by the exact td density functional 46 and pose a stringent test for approximations. In addition to these, are conditions that are particular to the time-dependent case, and involve aspects of memory-dependence. 5 In this section we analyze the approximations proposed in the previous section in the light of the fulfillment of the exact conditions enumerated below. This journal is the Owner Societies 2018 Phys. hem. hem. Phys., 2018, 20,

4 (i) Zero force theorem (ZFT) The zero force theorem ensures that the xc potential does not exert a net force, nr; tþrv xc ½nŠr; tþd 3 r ¼ 0: (9) Since the net force exerted by the Hartree potential vanishes, eqn (9) ensures that the inter-electron oulomb interaction doesn t exert any force on the system. Linear response (around the gs) applied on eqn (9) yields a link between the xc kernel f xc (r,r 0,o) and the gs xc potential v 0 xc(r) 51 that shows that frequency-dependence in a memory-dependent kernel is incompatible with a concurrent local-in-space density-dependence. (The Gross Kohn functional 52 thus violates the ZFT.) Violation of the ZFT has been shown to lead to numerical instabilities 53 due to the system self-exciting. (ii) Generalized translational invariance (GTI) Translational invariance requires the wavefunction in an accelerated frame (boost) to transform as, b Y N r 1...r N ; tþ ¼ e irj_btþ jr 1 þ btþ...r N þ btþ; tþi j¼1 (10) where b(t) is the position of the accelerated observer and b(0) = : b(0) = 0 such that the accelerated and inertial systems coincide at the initial time. The boosted density transports rigidly, n b (r,t) =n(r + b(t),t). (11) Vignale proved in ref. 50 that in order to fulfill GTI the xc potential must transform according to, v b xc[n;(0),f(0)](r,t) =v xc [n;(0),f(0)](r + b(t),t) (12) A v xc that fulfills eqn (12) automatically fulfills the ZFT eqn (9). 50 A special case of GTI is the harmonic potential theorem (HPT), which states that for a system confined by a harmonic potential and subject to a uniform time-dependent electric field, the density transforms rigidly following eqn (11) where b(t) is the position of the center of mass. 7,47,50,54 (It was shown that the Gross Kohn functional also violates GTI. 50 ) (iii) Memory condition The memory condition states that any instant in time can be regarded as the initial moment if the wavefunction is known at that time, and the xc potential should be invariant as to which previous time is used in its functional-dependence, 55 i.e. v xc [n t 0;(t 0 ),F(t 0 )]( - r,t) is independent of t 0 for t 4 t 0, (13) where n t 0(r,t) =n(r,t) for t 4 t 0 and is undefined for t o t 0. This is a very strict condition on the xc potential, tying together In current-density functional theory there is also a zero-torque version of this equation, but it does not hold generally in TDDFT where the Kohn Sham current might differ from the exact by a rotational component. initial-state and history dependences, and has implications also when the initial-state is a ground-state. (iv) 1-Electron self-interaction free (1e-SI-free) As in the ground-state case, the realization that an electron does not interact with itself yields the condition that for any one-electron system the correlation potential is zero and exchange cancels the Hartree potential, v 1e X r; tþ ¼ dr 0 nr 0 ; tþwjr r 0 jþ; v 1e r; tþ ¼0: (14) (v) onstant resonances condition (R) If a system in an arbitrary superposition state, tþ ¼ P n n e ient, evolving in a static Hamiltonian, is n probed to determine the excitation frequencies, o i, then the frequency predicted for a given transition should be independent of the state, i.e. of the n. Within TDDFT this condition requires a subtle cancellation between time-dependence of the xc kernel and the KS response function in a generalized non-equilibrium response formalism: 56,57 the exact (nonequilibrium) Hartree-xc kernel must compensate for timedependence in the poles of the KS response function in order to maintain constant o i. Given an approximation, this condition is difficult to test analytically; numerically it can be tested by turning off the external field at a given time T and kicking the system. The frequencies of the ensuing dipole moment should be independent of T if they satisfy this condition. Further, the fact that different choices of the KS initial state give different peak positions is also an indication of R violation. 57 Apart from the exact conditions examined here there are a number of other known exact conditions 46 that we have not yet studied in the context of these approxiations. For example, the behavior of the xc potential under uniform coordinate scaling, 58 which, most generally is connected with behavior under coupling-constant scaling, as in the ground-state. It is interesting to note that if the KS state was chosen as a Slater determinant, the exchange component scales simply in a linear fashion, however for more general KS states, the separation into exchange and correlation is no longer straightforward. 25 In any case, we leave to future work to reveal the behavior of the components v W xc and v T under uniform coordinate scaling, and coupling-constant dependence. A. Table of results. First, let us make some observations regarding the exact components of the xc potential; these are summarized in Table 1. All the proofs may be found in Appendix. The exact interaction and kinetic components, eqn (3) and (4), independently fulfill the ZFT, GTI, and memory condition and are 1e-SI-free. We can further break down the kinetic component v T into a part v T int that depends on the interacting 1RDM and another v T S that depends on the KS 1RDM: v T int corresponds to replacing Dr 1 - r 1 and v T S corresponds to replacing Dr 1 - r 1,S in eqn (4). So v T [n,dr 1 ] = v T int[n,r 1 ] v T S[n,r 1,S ]. The two components v T int and v T S fulfill Phys. hem. hem. Phys., 2018, 20, This journal is the Owner Societies 2018

5 Table 1 Fulfillment of exact conditions by the exact components of the xc potential Exact components ZFT GTI Memory cond 1e-SI-free R v W xc Yes Yes Yes Yes No v T = v T int v T S Yes Yes Yes Yes No v T int Yes No Yes No No v T S Yes No Yes No No the ZFT independently. GTI is only fulfilled by the difference of the two terms. We therefore expect a strong violation of GTI when the kinetic component is approximated by a term that asymmetrically approximates the interacting and KS 1RDMs (as in the case of v S xc þ vt;r 10Þ ). None of the individual components of the exact v xc satisfy independently the R in general; the full sum of these terms is generally required. In Table 2 we summarize the results for the approximations presented in Section IIA; the proofs can be found in Appendix for all conditions except the R. The latter is difficult to prove analytically, so we rely on numerical examples, and in all cases we found one example which explicitly showed peak shifting indicative of the violation of this condition. We separate the analysis of the v S xc component from the kinetic components, considering their satisfaction or violation of the exact conditions independently, although in practise we always evolve a kinetic component in conjunction with v S xc. The potential v S xc fulfills all exact conditions considered here, except for the R, and presents the more robust performance as we shall see in Section IV, although it does not always give results that are the ones closest to the exact. On the other hand, v T (0) violates all of them except 1e-SI-free. Freezing the difference in the 1RDMs, v T;Dr 10Þ fulfills the ZFT and is 1e-SI-free; it turns out to perform quite well for some of the dynamics when used in combination with v S xc as we will see in Section IV. However GTI is violated since the system does not transform properly under a boost, but the violation is likely weaker than that for v T;r 10Þ : there are large cancellations between v T int and v T S that cannot occur when freezing r 1 while evolving r S 1 (see discussion later). v T;Dr 10Þ also violates the memory condition and the R. On the other hand v T;r 10Þ fulfills the ZFT but, as mentioned before, violates GTI strongly due to the asymmetric treatment of the interacting and Kohn Sham 1RDMs. Further, this approximation violates the memory condition and is not 1e-SI-free. In Table 2 we also include the adiabatic local approximation (ALDA) and the adiabatically-exact (AE) approximation v AE xc ; the latter corresponds to propagating with the exact ground-state xc potential evaluated on the instantaneous density, v AE xc = v gs xc[n(t)]. v AE xc fulfills GTI and therefore also the ZFT and so does ALDA, 49,50,59 but ALDA is not 1e-SI-free. Because of the lack of dependence on the initial interacting and KS wavefunctions and on the history of the density, both adiabatic approximations trivially fulfill the memory condition. 55 But both ALDA and AE violate the R as was shown in ref. 56 and 57 respectively. IV. Analysis for non-equilibrium dynamics In this section we analyze the performance of the non-adiabatic approximations introduced in Section IIA to simulate electron dynamics. We focus on systems driven well beyond linear response from their gs, i.e. non-equilibrium situations, where we expect memory (non-adiabaticity) to be quite relevant. In general, the performance of an xc functional approximation will likely depend on the system, the particular dynamics under study, and the observables of interest; as evident from the literature, even the simplest approximation, ALDA, has a performance range of abysmal to accurate enough to be very useful. 5 This should be borne in mind when a limited range of studies is done, as here, however at the same time, these studies could be carefully used to indicate expected behavior of the functionals in more general situations. We choose a numerically exactly-solvable model system for which we can compute the exact evolution of the many-body wavefunction. We compare not only the performance of the approximation to simulate the density evolution but also the features of the potential as the dynamics evolves. The system is a one-dimensional model of the helium atom: two soft- oulomb interacting electrons in a soft-oulomb well (atomic units are used throughout), ^H ¼ X i¼1;2! 1 d 2 2dx 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ v app x i ; tþ i x i2 þ 1 1 þ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: jx 1 x 2 j 2 þ1 (15) Table 2 Fulfillment of exact conditions by the proposed approximations to the interaction and kinetic components. Also included are the adiabatically-exact (AE) and ALDA approximations Approximations ZFT GTI Memory cond 1e-SI-free R v S xc Yes Yes Yes Yes No v T (0) No No No Yes No v T;Dr 10Þ Yes No No Yes No Yes No No No No n T;r 1 0Þ v AE xc Yes Yes Yes Yes No xc Yes Yes Yes No No v ALDA where v app (x,t) is an applied potential, mimicking a laser field. All computations were performed using an in-house code. The spatial length of the simulation box is 40 a.u. with spacing 0.1 a.u. and has absorbing boundaries; the time-step used was r0.02 a.u. A. Field-free evolution of a superposition state Our first case study is a superposition state evolving freely, i.e. v app (x,t) = 0. We prepare the He atom in a 50 : 50 This journal is the Owner Societies 2018 Phys. hem. hem. Phys., 2018, 20,

6 superposition of the ground and the first-excited singlet state and we let it evolve freely, 0Þ ¼p 1 ffiffi 0 þ 1 Þ; (16) 2 where i = i (x 1,x 2 ) denote the many-body eigenstates of Hamiltonian eqn (15) with v app (x,t) = 0. The frequency of the dynamics corresponds to the energy difference between the ground and the first excited state, o 0 = E 1 E 0 = a.u. (period T 0 =2p/o 0 = a.u). Our proposed approximate functionals depend on the choice for the KS initial state F(0). Note that this is a separate statement than simply the fact that different initial states yield different dynamics under the same potential: here the potential itself depends on the initial state. The theorems of TDDFT state that any KS initial state may be chosen provided it has the same initial density and initial first time-derivative of the density as the true interacting state. The exact xc potential is different for each choice, but yields the same density-dynamics for each choice. The potentials in an adiabatic approximation are invariant to the choice and yield different density-dynamics for each (and some investigations have been made whether one can judiciously pick an initial Kohn Sham state for a given physical state in which an adiabatic approximations performs best 25,38,60 ). Like the exact xc potential, our approximate potentials are sensitive to the choice of initial KS state, but unlike the exact xc potential, the dynamics differs greatly depending on this choice. We explore the following choices: (1) Single Slater Determinant (SSD): F(x 1,x 2,0) = j(x 1 )j(x 2 ) (17) p where jxþ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nx; 0Þ=2, with n(x,0) the initial density of the interacting system, (2) Non-interacting 50 : 50 superposition: F 50:50 0Þ ¼p 1 ffiffif 0 x 1 ; x 2 ÞþF 1 x 1 ; x 2 ÞÞ; (18) 2 where F 0 is the ground state and F 1 is the first non-interacting singlet single excitation F 1 x 1 ; x 2 Þ¼p 1 ffiffij 0 x 1 Þj 1 x 2 Þþj 1 x 1 Þj 0 x 2 ÞÞ; (19) 2 with j 0 and j 1 the ground and first excited orbitals of a noninteracting potential such that the one-body density of F 50:50 (0), n F 50:50 0Þ ¼ j j 0xÞj 2 þjj 1 xþj þ 2 p ffiffi 2 Re ½ j0 xþj 1 xþš ¼ nx; 0Þ. To obtain j 0 (x) and j 1 (x) we use an iterative procedure that targets the initial density. The procedure is close to the ones found in ref. 61 and 62. At every iteration the eigenstates j i (x) are found for a guess of the potential v S (x) and a density n(x) is obtained from F 0 or F 50:50 (0). Then a correction in the potential is added, proportional to n(x) n target (x). Here we actually use l(n(x) n target (x))/(n(x) +e) where l and e are small numerical parameters. The convergence criterion is the difference between the two densities. We compare the densities and potentials resulting from propagating these initial KS states under our approximations, with the exact ones. To find the exact time-dependent potential, a variant of the global fixed point iteration method of ref. 63 and 64 was used. While the method of ref. 63 has been designed to also have a high accuracy even in the low density regions, which requires a careful treatment of the boundary conditions and low density regions, we are not so much focused on low densities and tail regions in this work. We therefore adopted a more pragmatic approach, introducing a cut-off that flattens the potential in the small-density regions. The cut-off takes the form gxþ ¼ 1 2 þ 1 tanh½anxþ cþš where c is the cut-off and a is a large 2 constant. For a potential of the form vyþ ¼ Ð y f xþdx applying the cut-off corresponds to vyþ ¼ Ð y gxþf xþdx. Since the method of ref. 63 in each time-step tries to get the right density, and thus within each time-step to compensate for the error in the density caused by for example the cut-off, it causes an overly oscillatory potential in the low density regions. We therefore further limited the number of iterations to three to avoid this issue. As an initial guess we used the analytic formulas, eqn (3) and (4), since we also propagate the correlated wave function. (Though analytically exact, these formulas alone do usually not lead to a stable numerical propagation: the numerical error created by the multiple derivatives and integrations leads to a density that drifts away from the reference density.) 1. SSD choice for U(0). In Fig. 1 we show the dipole dtþ ¼ Ð Ð xnx; tþdx and norm NtÞ ¼ nx; tþdx for the field-free evolution of the superposition state of eqn (16) with the first KS initial state choice of the previous subsection, the SSD eqn (17). The exact dynamics is shown by the black line. Propagating using v S xc (in green dashed) which, for two electrons and the SSD choice, corresponds to adiabatic exact-exchange (AEXX) yields a dipole that appears to oscillate with more than one frequency, giving an envelope to the oscillations, and neither the dominant frequency nor the amplitude of oscillations are captured well. The ALDA propagation (blue dotted) is also shown, and it approximates the frequency of the oscillation a little better than AEXX, but the amplitude is poor and again there is a beating over long time. The propagations that include the various approximate v T improve over the traditional approximations for only a very short time but soon deviate dramatically with qualitatively wrong behavior even before half a cycle of the evolution. We will now discuss what goes wrong with each of these, with the help of the density and potential time-snapshots within the first optical cycle shown in Fig. 2. We first note that when a SSD is chosen for the KS system, with one doubly-occupied orbital, the features of the exact v xc (black line) have the periodicity of the density dynamics: initially the density has a minimum on the left side coincident with a large dynamical step and peak in v T, which decreases and disappears as the density becomes symmetric around t = 3 a.u., before building up again on the other side, such that at half-cycle (t E 6a.u.)v xc and the density are mirror images of their initial values Phys. hem. hem. Phys., 2018, 20, This journal is the Owner Societies 2018

7 Fig. 1 Field-free dynamics of the 50 : 50 superposition state eqn (16) where the approximate TDDFT calculations propagate the SSD choice for initial KS state F(0), eqn (17). Norm (upper panel) and dipole moment (lower panel) for exact (black line), v S xc = AEXX (green dashed), v S xc + v T (0) (magenta triangles), v S xc þ vt;dr 1 0Þ squares) and v ALDA xc following graphs. (red crosses), v S xc þ vt;r 1 0Þ (light blue (blue dotted). Time is given in a.u. here and in all onsider now the potentials of the traditional approximations, v S xc = AEXX and ALDA (in green dashed and blue dotted in Fig. 1 and 2). Although both the AEXX and ALDA dipoles are close to the exact for t o 3 a.u. the shape of the density at this time has already started to deviate. The v S xc potential is generally a very good approximation to the exact interaction part v W xc of the xc potential 25 but it completely misses the step and peak features present in the exact v T, simply smoothly cradling the density as it evolves, and ALDA also does not capture these structures. These missing structures are important to get the details of the dynamics right, including the period, as evident in the plots. onsider now propagation with v S xc + v T (0) (magenta triangle curves in Fig. 1 and 2). As v T (0) is static, the large initial step in the kinetic correlation part of the potential remains for all time, and as a consequence the density leaks to the left and is unable to slosh back to the right (see the magenta triangle density in the left panel of Fig. 2). The density eventually gets absorbed by the boundary around t E 30 a.u. as can be seen in the evolution of the norm in the upper panel of Fig. 1. We turn now to propagation under v S xc þ vt;dr 1 0Þ, where again the dipole swings far too much to the left (red crosses in Fig. 1), resulting in a large unphysical absorption evident in the norm in the top panel. The reason for this becomes evident from the right panel of Fig. 2: freezing Dr 1 in v T results in a step that, although varying in size, is always on the left, pulling the density consistently over to the left. On the other hand, propagation under v S xc þ vt;r 1 0Þ yields a dipole that displays a most peculiar flattening after half a cycle (light blue squares in Fig. 1 and 2). The freezing of the interacting 1RDM while keeping the KS 1RDM dynamical has drastic consequences, as can be seen from first recalling the exact v T : both the exact v T int and v T S display large dynamical steps that tend to counteract each other, yielding a smaller step in Fig. 2 Density (upper panel) and v xc potential (lower panel) for the same dynamics as Fig. 1 shown at initial time, and at three other times indicated within the first cycle. Left panel: Exact (black line), v S xc = AEXX (green dashed), v S xc þ vt 0Þ (magenta triangles) and valda xc (blue dotted). Right panel: Exact (black line), v S xc þ vt;dr 1 0Þ (red crosses) and v S xc þ vt;r 1 0Þ (light blue squares). The v S xc þ vt;r 1 0Þ potential is scaled by a factor of 10 5 at t = 6 a.u. and t = 9 a.u. the exact v T, at times even canceling each other (see Fig. 3 and Movie 1 in ESI ). Freezing only the interacting 1RDM as in v T;r 1 0Þ leads to the development of a disproportionately large step at times, resulting in instabilities and the poor dynamics as shown here. Satisfaction of the ZFT is demonstrated in Fig. 4 which plots the left-hand-side of eqn (9) for the different approximations. From Table 2, all approximations except for v S xc + v T (0) are expected to satisfy the ZFT, and give zero. The deviation of v S xc þ vt;r 10Þ (light blue squares) is a numerical error related to the very large steps that appear in this approximation as just discussed. The spike in v S xc þ vt;dr 1 0Þ (red crosses) at around 25 a.u. is a numerical artifact caused by the step in the potential at that time being very large and sharp. Notice the relatively large violation of the ZFT for v S xc + v T (0) (magenta triangles) in this dynamics. We conclude that except for quite short times, the new approximations do not improve the dynamics of a field-free superposition state when the initial KS state is chosen to be a SSD; in fact the traditional approximations ALDA and AEXX This journal is the Owner Societies 2018 Phys. hem. hem. Phys., 2018, 20,

8 Fig. 3 Exact components v T int (red dashed) and v T S (blue dotted) of the exact kinetic component v T = v T int v T S (black thick line) for field-free dynamics evolution of superposition state eqn (16) at initial time and at about 1/4 cycle (t = 3 a.u.). Top panel: SSD choice eqn (17) for the KS initial state. Lower panel: Superposition choice eqn (18) for the KS initial state. Fig. 4 Violation of the ZFT: the left-hand-side of eqn (9) in fieldfree evolution of superposition state eqn (16), in propagation under v S xc (green dashed), v S xc + v T (0) (magenta triangles), v S xc þ vt;dr 1 0Þ (red crosses), v S xc þ vt;r 1 0Þ (light blue squares) and v ALDA xc (blue dotted). Upper panel: SSD choice eqn (17) for the KS initial state. Lower panel: Superposition choice eqn (18) for the KS initial state. (coinciding with v S xc in this case) are more reliable, although inaccurate. One might be tempted to argue that these results have limited relevance, since starting the KS system in an SSD when the interacting state is a 50 : 50 superposition state would be a bad choice, making a challenging job for approximate functionals to do well. However, this could well be the situation reached when the system began in the ground-state, with some field applied that brought the interacting state to a 50 : 50 superposition state, at which point the field is turned off. Then the natural choice for the initial KS state is the KS ground-state, and we are stuck with a SSD for the whole evolution. 2. Superposition choice for U(0). Fig. 5 shows the field-free superposition state dynamics for the different approximations Fig. 5 Time-dependent dipole moment (lower panel) and norm (top panel) for field-free dynamics of 50 : 50 superposition state eqn (16). Initial KS state F(0) is chosen as in eqn (18). Shown are the results of propagation under the exact (black line), v S xc (green dashed), v S xc + v T (0) (magenta triangles), v S xc þ vt;dr 1 0Þ (red crosses), v S xc þ vt;r 1 0Þ (light blue squares), AEXX ¼ v H (yellow doubled dot dashed) and ALDA (blue dotted). 2 when the KS initial state is chosen to be eqn (18). onsidering first the traditional approximations ALDA and AEXX = v H /2, we observe a great improvement in their performance with this choice of KS initial state, compared to the Slater determinant choice. In fact ALDA is remarkably good: choosing a KS initial state structure that is close to that of the true system seems to override the errors from the ground-state xc assumption inherent in the adiabatic nature of ALDA, its self-interaction error, and its locality in space. It is the configuration of the initial KS state that is key: if instead we used LDA orbitals but still within the first-excited Slater determinant configuration of eqn (18), the dynamics is similar although not as accurate (see shortly). In fact ALDA outperforms the spatially non-local AEXX, suggesting the importance of an adequate accounting of correlation for this example, with even the local-density ground-state correlation approximation doing a good job here. We will see that freezing the kinetic component of the correlation in various ways in our new approximations, together with the correlation contained in v S xc, does not provide a sufficiently good correlation component to compete with the simple ALDA for more than short to intermediate times. Turning then to the new approximations, we first note that, unlike the SSD choice, none of the approximations lead to the unphysical situation of the density reaching the boundaries and being absorbed, at least until much later (for v S xc + v T (0), magenta triangles), as shown by the norm. Two of the approximations, namely v S xc (which gives dynamics close to AEXX) and v S xc þ vt;dr 10Þ, do a reasonable job however none of them beats the remarkable performance of ALDA for the dipole dynamics as noted above. Inclusion of v T;Dr 1 0Þ tends to yield a worse amplitude but a better dominant frequency than v S xc. Performing a Fourier transform of the dipole yields the frequencies o = a.u. for v S xc þ vt;dr 10Þ as compared with o = a.u. for v S xc while the exact frequency is Phys. hem. hem. Phys., 2018, 20, This journal is the Owner Societies 2018

9 ESI displays the densities and xc potentials from exact propat;dr 0Þ gation, ALDA, vsxc, and vsxc þ v 1. Turning now to the other two approximations, vsxc + vt(0) and T;r 0Þ vsxc þ v 1, the performances are poor, although the release of density then absorbed by the boundaries is less compared to the SSD case. This is because the step in the initial kinetic components is less, so even if frozen as in vsxc + vt(0), it results in less density moving out to the left. Propagation with T;r 0Þ vsxc þ v 1 (light blue squares) suffers from the same problem as in the SSD choice: the uneven treatment of the interacting and KS density matrices results in a huge step in the potential that prevents the density from sloshing back and forth as it should (right panel of Fig. 6). Finally, since typically orbitals that reproduce the exact density are not easily obtained, we show in Fig. 7 the results of propagating a state of the form eqn (18) where the LDA ground and first-excited orbitals are inserted into the KS states. We show the best of our new approximations, along with ALDA. We notice that the density-dynamics is generally significantly T;Dr 0Þ worsened for both propagation with ALDA and vsxc þ v 1 compared to when initial orbitals that reproduce the exact initial density are used. B. Fig. 6 Time snap-shots of the density (upper panels) and vxc potential (lower panels) for the dynamics shown in Fig. 5. Left panel: Exact (black line), vsxc (green dashed), vsxc + vt(0) (magenta triangles) and valda (blue xc T;Dr 0Þ dotted). Right panel: Exact (black line), vsx þ v 1 (red crosses), and T;r 0Þ vsxc þ v 1 (light blue squares). The latter has been scaled by 10 4 for t = 9 a.u. and t = 15 a.u. Arbitrary field starting in the ground state Next we study the dynamics of system eqn (15) initially in the gs and driven by a non-resonant, fairly strong, laser field, vapp(x,t) = E(t)x, where E(t) = 0.1 sin (0.4t). Since we start in the gs the natural choice is a SSD for the KS initial state, and this is the only choice we consider here. In Fig. 8, the dipoles and norm for the exact and o0 = a.u., best approximated by ALDA s frequency of o = a.u. T;Dr 0Þ These two new approximations, vsxc and vsxc þ v 1, yield a good density evolution, as can be seen in Fig. 6 in green dashed and red crosses respectively, where they can be compared with the exact density and potential (black line). The exact vt for this choice of KS initial state is initially much smaller than for the SSD choice (cf. Fig. 2), and vsxc, which, as we noted before is a very good approximation to vw xc, then plays a relatively more important role than in the SSD case. We note that vsxc in this case is not equivalent to TDEXX: it contains some correlation due to the KS state configuration, and has memory-dependence in that it is an implicit functional of the density at earlier times T;Dr 0Þ and of the KS initial state. Including v 1 now partially captures the step and peak features of the exact potential and the step switches sides unlike for the SSD case. Notice that from approximately t = 6 a.u. onwards the exact vxc is no longer periodic while the density is (unlike the case in the previous section where a SSD is used). The dynamical features of the exact vt become more and more complex as the system evolves and at later times they become eventually as large as the ones observed for the SSD choice (see also ref. 25); the Movie S2 in This journal is the Owner Societies 2018 Fig. 7 Time snap-shots of the density (upper panel) and vxc potential (lower panel) for field-free dynamics starting in 50 : 50 superposition state eqn (16) and using a KS initial state of the form eqn (18) but with the gs and T;Dr 0Þ first excited LDA orbitals. Propagation with vsxc þ v 1 (red crosses) and ALDA vxc (blue dotted). In the black line the exact dynamics. Phys. hem. hem. Phys., 2018, 20,

10 Fig. 8 Dynamics of the laser-driven gs. Norm (upper panel) and dipole (lower panel) for exact (black line), v S xc = AEXX (green dashed), v S xc þ vt;dr 1 0Þ (red crosses), v S xc + v T (0) (magenta triangles), v S xc þ vt;r 1 0Þ (light blue squares) and v ALDA xc (blue dotted). approximate KS evolutions, beginning in the initially exact KS rffiffiffiffi n 0 orbital j0þ ¼,areshown. 2 The propagation with v S xc = AEXX alone (in green dashed) is excellent till almost t = 12 a.u., after which it begins to significantly deviate from the exact. The other traditional approximation ALDA (in blue dotted) is not as good as AEXX in this case for the first few cycles but stays closer to the exact than AEXX at later times. Fig. 9 shows that in fact both these approximations and the new ones, except for v S xc þ vt;r 1 0Þ which is not shown (see below), capture the structure of the density and the potential quite well in the central region; the noticeable deviations of the dipole moments of the approximations in Fig. 8 arise from errors in the density further out where the density itself is not very large, but these errors get enhanced by the multiplication by x in the calculation of the dipole moment (which, additionally, highlights asymmetric structures in the density). Adding the correlation potential frozen to its initial value, v S xc + v T (0) (in magenta triangles), does not improve the results over v S xc (except at short times) as can be seen by comparing the green dashed and magenta triangle curves in Fig. 8 and in the density time snapshots of Fig. 9. In this case the system starts off weakly correlated, and the exact potential is a smooth well initially, but as the laser field kicks in and begins to drive the interacting system far away from approximately a single Slater determinant, correlation increases, as is reflected in the increased size of the structures in the exact v T evident in Fig. 9; these structures appear in v T rather than in v W xc which remains quite smooth. As v S xc + v T (0) freezes the v T to its initial value, it cannot capture these. The approximation v T;Dr 10Þ (red crosses) does include partial step structures, although not always quite in phase with the exact (see also Movie S3 in the ESI ). However, these do not apparently have a significant effect on the ensuing dynamics, neither in the region where the density is appreciable, nor in the dipole moment that weighs more the low-density regions Fig. 9 Density (upper panels) and v xc potential (lower panels) for the laser-driven gs. Left panel: Exact (black line), v S xc = AEXX (green dashed) and v S xc + v T (0) (magenta triangles). Right panel: Exact (black line), v S xc þ vt;dr 1 0Þ (red crosses) and v ALDA xc (blue dotted). further out, and the dipole moment is not better than the traditional approximations except at short times. One aspect of the largest of these step structures for this dynamics, is that they can change very rapidly, for example the large step switches sides from t = 43 a.u. to 44 a.u. Also, at t E 60 a.u. the exact dynamics begins to show some absorption (see upper panel Fig. 8) which is underestimated in v S xc þ vt;dr 1 0Þ. This leads to some discrepancy in the dipole moments, but the density in the central region is very well reproduced still (see the lower panels in red crosses in Fig. 9). The propagation under v S xc þ vt;r 1 0Þ develops very large features near the boundaries (no compensation of the large step in v S, as in the discussion in the previous section) that progressively approach the central region, growing in size, and by t E 5 a.u. the step has become so large and sharp, that the propagation results in a freezing of the density on the left (in light blue squares in Fig. 8). Finally, in practice we rarely have access to the exact initial KS orbitals, and approximate DFT orbitals are used for the initial state. In Fig. 10 LDA orbitals are propagated using v S xc þ vt;dr 10Þ and ALDA, showing a noticeable deviation from the results when the exact initial orbitals are used, especially, in this case for the propagation under v S xc þ vt;dr 1 0Þ Phys. hem. hem. Phys., 2018, 20, This journal is the Owner Societies 2018